Systems of Equations & Inequalities — Drills
Systems of Equations & Inequalities has 9 drill skills. Each one generates fresh questions every time, with a hint and full worked steps, so you can keep practising a single skill until it settles.
What these drills cover
These questions use 6 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Inequality — A statement that one expression is less than or greater than another.
- Perimeter — Distance all the way around the outside.
- Volume — Space inside, in cubic units.
- System — Two or more equations that must hold at the same time.
- Parallel — Same slope, different intercept.
- Boundary line — The line you would get by replacing the inequality sign with =.
Substitution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve y = 4x - 4 and y = 2x + 8.
Worked answer
(6, 20)
4x - 4 = 2x + 8
2x = 12
x = 6, so y = 20
Problem 2
Solve y = 4x + 6 and y = 1x - 9.
Worked answer
(-5, -14)
4x + 6 = 1x - 9
3x = -15
x = -5, so y = -14
Elimination
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve 5x + 7y = -13 and 5x - 7y = -27.
Worked answer
(-4, 1)
Add: 10x = -40
x = -4
y = 1
Problem 2
Solve 1x + 5y = -5 and 1x - 5y = 5.
Worked answer
(0, -1)
Add: 2x = 0
x = 0
y = -1
Checking a solution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Is (2, 4) a solution of 5x + 1y = 14 and x - y = -2? Answer yes or no.
Worked answer
yes
First: 5(2) + 1(4) = 14, needed 14
Second: 2 - (4) = -2, needed -2
Both pass, so yes.
Problem 2
Is (5, 4) a solution of 3x + 5y = 20 and x - y = 4? Answer yes or no.
Worked answer
no
First: 3(5) + 5(4) = 35, needed 20
Second: 5 - (4) = 1, needed 4
At least one fails, so no.
Boundary
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y ≥ 2x + 1, is the boundary line solid or dashed?
Worked answer
solid
The ≥ includes the line.
So solid.
Problem 2
For y > 2x + 1, is the boundary line solid or dashed?
Worked answer
dashed
A strict inequality excludes the line.
So dashed.
Multiplying
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
Solve x + 5y = 12 and 3x - y = 4.
Worked answer
(2, 2)
× 5: 15x - 5y = 20
Add: 16x = 32
x = 2, y = 2
Problem 2
Solve x + 3y = 5 and 2x - y = 10.
Worked answer
(5, 0)
× 3: 6x - 3y = 30
Add: 7x = 35
x = 5, y = 0
Shade
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
For y ≤ 2x - 7, does the shading include (0, 0)? Answer yes or no.
Worked answer
no
At (0,0): is 0 ≤ -7?
False, so the origin is outside it.
Problem 2
For y ≥ 3x + 5, does the shading include (0, 0)? Answer yes or no.
Worked answer
no
At (0,0): is 0 ≥ 5?
False, so the origin is outside it.
No solution
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
How many solutions does 6x - 3y = 5 and 12x - 6y = 12 have?
Worked answer
none
× 2: 12x - 6y = 10
But the second says 12.
Parallel lines, so no solution.
Problem 2
How many solutions does 2x - 3y = 10 and 4x - 6y = 24 have?
Worked answer
none
× 2: 4x - 6y = 20
But the second says 24.
Parallel lines, so no solution.
Word problems
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A shop sells $11 and $18 notebooks. It sold 20 notebooks for $311. How many of the cheaper kind?
Worked answer
7
a + b = 20 and 11a + 18b = 311
b = 20 - a, so 11a + 18(20 - a) = 311
-7a = -49
a = 7
Problem 2
A shop sells $3 and $9 notebooks. It sold 25 notebooks for $129. How many of the cheaper kind?
Worked answer
16
a + b = 25 and 3a + 9b = 129
b = 25 - a, so 3a + 9(25 - a) = 129
-6a = -96
a = 16
From a graph
Unlimited questions of this kind are generated in the app. Two examples:
Problem 1
A system of two linear inequalities has a wedge-shaped solution region. This means
- the two boundary lines cross
- the lines are parallel
- the lines are identical
- there is no solution
Worked answer
the two boundary lines cross
Two crossing lines cut the plane into four wedges.
The solution is whichever wedge satisfies both.